1. Field of the Invention
The present invention relates to a resonator for outputting a resonant frequency signal.
2. Description of the Related Art
In recent years, MEMSs (Micro Electro Mechanical Systems), which are extremely small devices where an electromechanical mechanism is formed on a substrate made of, e.g., Si by using ultrafine processing to have mechanical and electrical functions, have been attracting attention as enabling the realization of functions that have not been realized. An electromechanical mechanism formed on a substrate includes a mechanical element, a sensor, an actuator, and an electronic circuit, etc., integrated on a substrate.
Actuators used in MEMSs include ones using an electrostatic force, ones using an electromagnetic force, ones using a piezoelectric effect, ones using thermal distortion, etc. In particular, actuators using an electrostatic force as driving means are most often used which are simple in structure and easy to miniaturize. Moreover, a parallel plate-type electrostatic actuator, a comb-teeth-shaped electrostatic actuator, an electrostatic motor, or the like is used as an actuator using an electrostatic force as driving means.
Commercially available products actually using an MEMS include inkjet printers, pressure sensors, acceleration sensors, gyroscopes, DMDs (projectors), etc. It is possible to produce resonators by the MEMS technology, and because the resonators are small-sized and can realize a high Q value, they can be used instead of crystal filters and crystal oscillators.
For the above resonators, the bias voltage applied between their input and output and the resonant frequency are important, and for parallel plate-type resonators, the following equation 1, a relational expression, is satisfied. Refer to Equation 12 in F. T. Bannon, J. R. Clark, C. T.-C. Neuyen, “High-Q HF Microelectromechanical Filters” proceedings, IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 35M No. 4, APRIL 2000, pp. 512-526.
                    f        =                                            1                              2                ⁢                                                                  ⁢                π                                      ⁢                                                                                k                    m                                    -                                      k                    e                                                  m                                              =                                    1                              2                ⁢                                                                  ⁢                π                                      ⁢                                                            k                  m                                m                                      ⁢                                          (                                  1                  -                                                            k                      e                                                              k                      m                                                                      )                                            1                2                                                                        (                  Equation          ⁢                                          ⁢          1                )            
Here, m is the mass of the resonator, ke is the electrical spring constant of the resonator, and km is the mechanical spring constant of the resonator. Generally, the equation 2 is satisfied and hence the equation 1 can be Taylor expanded into the equation 3.
                              k          m                >>                  k          e                                    (                  Equation          ⁢                                          ⁢          2                )                                f        =                              1                          2              ⁢                                                          ⁢              π                                ⁢                                                    k                m                            m                                ⁢                      (                          1              -                                                k                  e                                                  2                  ⁢                                                                          ⁢                                      k                    m                                                                        )                                              (                  Equation          ⁢                                          ⁢          3                )            
The electrical spring constant ke of the resonator is given by the equation 4.
                              k          e                =                              ɛ            ⁢                                                  ⁢            A            ⁢                                                  ⁢                          V              2                                            d            3                                              (                  Equation          ⁢                                          ⁢          4                )            
Substituting the equation 4 into the equation 3, the equation 5 is obtained.
                                                        f              =                                                1                                      2                    ⁢                                                                                  ⁢                    π                                                  ⁢                                                                            k                      m                                        m                                                  ⁢                                  (                                      1                    -                                                                  ɛ                        ⁢                                                                                                  ⁢                        A                        ⁢                                                                                                  ⁢                                                  V                          2                                                                                            2                        ⁢                                                                                                  ⁢                                                  k                          m                                                ⁢                                                  d                          3                                                                                                      )                                                                                                        =                                                                    -                                                                  ɛ                        ⁢                                                                                                  ⁢                        A                                                                    2                        ⁢                                                                                                  ⁢                        π                        ⁢                                                                                                  ⁢                                                  k                          m                                                ⁢                                                  d                          3                                                                                                      ⁢                                                                                    k                        m                                            m                                                        ⁢                                      V                    2                                                  +                                                      1                                          2                      ⁢                                                                                          ⁢                      π                                                        ⁢                                                                                    k                        m                                            m                                                                                                                              (                  Equation          ⁢                                          ⁢          5                )            
Here, A is the opposite area of the opposite electrodes of the resonator, V is the bias voltage between the electrodes of the resonator, d is the distance between the electrodes of the resonator, and ∈ is the permittivity of the dielectric material.
From the equation 5 it is seen that the resonant frequency f is proportional to −V2. Further, since the value indicating the magnitude of bias voltage dependency is the coefficient of V2 in the equation 5, the value is determined by the electrode opposite area A, the electrode-to-electrode distance d, the permittivity ∈ of the dielectric material, the mass m of the resonator, and the mechanical spring constant km of the resonator.
Meanwhile, as a resonator with comb-teeth-shaped electrodes, there is known a surface acoustic wave device where an IDT for SAW resonance and a pair of distributed reflectors are formed on a crystal substrate as described in Japanese Patent Kokai No. H06-13836.